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HW8 - Physics2210.Homeworkassignment8 B 7.9.10 f/2 0/2 x...

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1 Physics 2210. Homework assignment 8. B 7.9.10 f(x)=|x| is an even function and has a period T= π for this problem. The Fourier series should be f(x) = a 0 2 + a n cos( 2n π x T ) n = 1 = a 0 2 + a n cos(2nx) n = 1 . a 0 = 4 π f(x)dx 0 π /2 = 4 π xdx 0 π /2 = π 2 , a n = 4 π f(x)cos(2nx)dx 0 π /2 = 4 π xcos(2nx)dx 0 π /2 = 1 π n 2 [( 1) n 1] , for n=1,2,… f(x) = π 4 + 1 π 1 n 2 [(-1) n -1]cos(2nx) n = 1 = π 4 2 π 1 (2n -1) 2 cos[2(2n -1)x] n = 1 . T&M 3.28 f(t) is an odd function and has a period T=2 π / ω for this problem. The Fourier series should be f(t) = b n sin( 2n π t T ) n = 1 = b n sin(n ω t) n = 1 . b n = 2 ω π f(t)sin(n ω t)dt 0 π / ω = 2 ω π sin(n ω t)dt 0 π / ω = 2 π n [( 1) n 1] , for n=1,2,… f(t) = 2 π 1 n [1- (-1) n ]sin(n ω t) n = 1 = 4 π 1 (2n -1) sin[(2n -1) ω t] n = 1 . f x π /2 0 /2 - 2.0 - 1.0 0.0 1.0 2.0 f(t) - 1.0 - 0.5 0.0 0.5 1.0 t/ π
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2 T&M 3.38 F(t) = 0, for t < 0 F 0 e γ t sin ω t, for t > 0 . With the Green’s function method, the solution for the oscillator’s response is x(t) = F(t')G(t,t')dt' 0 t ,
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